Model, Politics

The Graph Model of Conflict Resolution – Sensitivity Analysis

[10 minute read]

Last week, I used the Graph Model of Conflict Resolution to find a set of stable equilibria in the present conflict between North Korea and the USA. They were:

  • The tense status quo (s. 0)
  • An American troop withdrawal, paired with North Korea giving up its nuclear weapons (s.10)
  • All out conventional warfare on the Korean Peninsula (s. 4)
  • All out nuclear warfare on the Korean Peninsula (s. 5)

But how much can we trust these results? How much to they depend on my subjective ranking of the belligerent’s preferences? How much do they depend on the stability metrics I used?

To get a sense of this, I’m going to add another stability metric into the mix, come up with three new preference vectors, and look at how the original results change when we consider a North Korean invasion to be irreversible. After these eight new stability calculations, we’ll have nine slightly different ways of looking at the conflict; this should help us guess which equilibria are robust to my subjective choices and which might exist only because of how I framed the problem.

Alternative Stability Metrics

Previously we assessed stable states using Nash Stability and Sequential Stability. Sequential Stability allowed us to see what would happen if the decision makers were looking two moves ahead and assuming that their opponents wouldn’t “cut off the nose to spite the face” – it assumes, in essence, that people will only sanction by moving to states that they like more, not states they like less.

Maybe that’s a bad assumption dealing with Trump and Kim Jong-un. In this case, wouldn’t it be better to use Symmetric Metarationality? With Symmetric Metarationality, all sanctioning unilateral moves are on the table. Symmetric Metarationality also allows decision makers to respond to sanctioning. In effect, it lets them look three moves ahead, instead of the two allowed by Sequential Stability.

Before we see how this new metric changes things, let’s review our states, preference vectors, and stability analysis from last time.

The states are:

Click for a copyable version

Or in plain English:

State Explanation
0 Status quo
1 Nuclear strike by the US, NK keeps nuclear weapons
2 Unilateral US troop withdrawal
4 North Korean invasion with only conventional US responses
5 North Korean invasion with US nuclear strike
6 US withdrawal and North Korean Invasion
8 Unilateral North Korean abandonment of nuclear weapons
9 US strike and North Korean abandonment of nuclear weapons
10 Coordinated US withdrawal and NK abandonment of nuclear weapons
12 NK invasion after abandoning nuclear weapons; conventional US response
13 NK invasion after abandoning nuclear weapons; US nuclear strike
14 US withdrawal paired with NK nuclear weapons abandonment and invasion

From these states, we saw the following equilibria and unilateral improvements:

Click for copyable version

When dealing with Symmetric Metarationality, I find it very helpful to modify the chart above so that it also includes unilateral moves. After we make this change and blank out our results, we get the following:
Click for copyable version

From here, we use a simple algorithm. First, all states without unilateral improvements are Nash Stable. Next, we check each unilateral improvement in the remaining states against the opponent’s unilateral actions, then against the original actors best unilateral action from each of the resulting states. If there are no results lower than the original actor started, the move is unstable. Otherwise it’s stable by Symmetric Metarationality (and we’ll mark it with “S”). Like Sequential Stability, you can’t truly call this done until you check for states that are simultaneously sanctioned (this is often easy because simultaneous sanctioning is only a risk when both sides are unstable).

An example: There exist a unilateral improvement for America from s. 4 to s. 5. From s. 5, North Korea can move to s. 1, 13, or 9. America disprefers both s. 1 and s. 13 to s. 4 and has no moves out of them, so the threat of North Korea taking either of those actions is an effective sanction and makes s. 4 stable on the American side.

Once we repeat this for all states across both sides, we get the following:

Click for copyable version

We’ve kept all of our old equilibria and gained a new one in s. 12: “NK invasion after abandoning nuclear weapons; conventional US response”.

Previously, s. 12 wasn’t stable because North Korea preferred the status quo (s. 0) to it and the US had no UIs from the status quo. North Korea moving from s. 12 to s. 0 is sanctioned in Symmetric Metarationality by the US unilateral move from s. 0 to s. 1, which leaves North Korea with only the option of moving from s. 1 to s. 5. State 5 is dispreferred to s. 12 by North Korea, so it can’t risk leaving s. 12 for s. 0. State 12 was always Nash Stable for the US, so it becoming stable for North Korea makes it an equilibrium point.

To put this another way (and to put an example on what I said above), using Symmetric Metarationality allows us to model a world where the adversaries see each other as less rational and more spiteful. In this world. NK doesn’t trust the US to remain at s. 0 if it were to call for a truce after an invasion, so any invasion that starts doesn’t really end.

It was heartening to see all of our existing equilibria remain where they were. Note that I did all of the work in this post without knowing what the results would be and fully prepared to publish even if my initial equilibria never turned up again; that they showed up here made me somewhat relieved.

Irreversible Invasions

Previously we modelled invasions as reversible. But is this a realistic assumption? It’s very possible that the bad will from an invasion could last for quite a while, making other strategies very difficult to try out. It’s also likely that America wouldn’t just let North Korean troops give up and slink away without reprisal. If this is the case, maybe we should model a North Korean invasion as irreversible. This will mean that there can be no unilateral improvements for North Korea from s. 4, 5, or 6 to s. 0, 1, 2, 8, 9, or 10.

In practical terms, modelling an invasion as irreversible costs North Korea one unilateral improvement, from s. 4 to s. 0. Let’s see if this changes the results at all (we’re back to sequential stability):

Click for copyable version

We end up losing the simultaneous sanctioning that made s. 4 a stable state, leaving us with only three stable states: the status quo, a trade of American withdrawal for the North Korean nuclear program, and all out nuclear war on the Korean Peninsula.

We’ve now tried three different ways of looking at this problem. Three equilibria (s. 0, 10, 5) showed up in all cases, one in two cases (s. 4), and one in one case (s. 12). We’re starting to get a sense for which equilibria are particularly stable and which are more liable to only pop up under certain conditions. But how will our equilibria fare when faced with a different preference vectors?

Bloodthirsty Belligerents

What if we’ve underestimated how much North Korea and the United States care about getting what they want and overestimated how much they care about looking reasonable? I’m going to try ranking the states so that North Korea always prefers invading and the US always prefers first that North Korea doesn’t invade the South and second that they have no nuclear weapons program.

This gives us the following preference vectors:

US: 8, 9, 0, 10, 13, 12, 5, 4, 1, 2, 14, 6
NK: 6, 14, 4, 12, 5, 13, 2, 0, 10, 1, 9, 8

Since we’re modelling the actors as more belligerent, let’s also assume for the purposes of these analyses that invasions are irreversible.

Here are the preferences vectors we’ll use to find equilibria:

Click for copyable version

Sequential Stability

Click for copyable version

Here we have only two stable states, s. 5 and 12. Both of these involve war on the Korean Peninsula; not even the status quo is stable. State 2 is at risk of simultaneous sanctioning, but the resulting states (4, 12, 5, 13) aren’t dispreferred, to s. 2 for either actor, so no simultaneous sanctioning occurs. There really are just two equilibria.

Symmetric Metarationality

Click for copyable version

Symmetric Metarationality gives us the exact same result. Only s. 5 and s. 12 are stable. This is suspicious, as the conflict has managed to stay in s. 0 for quite some time. If these preferences were correct, North Korea would have already invaded South Korea and been met with a nuclear response.

What if these preferences are substantially correct and both sides are more aggressive than we initially suspected, but North Korea disprefers being attacked by nuclear weapons below s. 0 and s. 10? That state of affairs is perhaps more reasonable than the blatantly suicidal North Korea we just imagined. How does a modicum of self-preservation change the results?

Nuclear Deterrence

If we’re assuming that North Korea has broadly similar preferences to our last variation, but doesn’t want to get attacked by nuclear weapons, we get the following preference vectors:

US: 8, 9, 0, 10, 13, 12, 5, 4, 1, 2, 14, 6
NK: 6, 14, 4, 12, 0, 10, 5, 13, 2, 1, 9, 8

Here are the annotated preferences vectors we’ll use to assess stability with Sequential Stability and Symmetric Metarationality. Since we’re leaving the belligerency of the United States the same, we’ll continue to view invading as an irreversible action.

Click for copyable version

Sequential Stability

Click for copyable version

One “minor” change – deciding that North Korea really doesn’t want to be nuked – and we again have the status quo and a negotiated settlement (in addition to two types of war) as stable equilibria. Does this hold when we’re using Symmetric Metarationality?

Symmetric Metarationality

Click for copyable version

Again, we have s. 0, 5, 10, and 12 as our equilibria.

As we’ve seen throughout, Symmetric Metarationality tends to give very similar answers to Sequential Stability. It’s still worth doing – it helps reassure us that our results are robust, but I hope by now you’re beginning to see why I could feel comfortable making an initial analysis based just off of just Sequential Stability.

Pacifistic People

What instead of underestimating the bloodthirstiness of our belligerents, we’ve been overestimating it? It’s entirely possible that both sides strongly disprefer all options that involve violence (and the more violence an option involves, the more they disprefer it) but talk up their position in hopes of receiving concessions. In this case, let’s give our actors these preference vectors:

US: 8, 0, 10, 2, 9, 12, 4, 5, 14, 13, 6, 1
NK: 6, 14, 2, 10, 0, 8, 4, 12, 5, 1, 9, 13

(Note that I’m only extending “peacefulness” to these two actors; I’m assuming that North Korea would happily try and annex South Korea if there was no need to fight America to do so)

There are fewer unilateral improvements in this array than in many of the previous ones.

Click for copyable version

Sequential Stability

Click for copyable version

This is perhaps the most surprising result we’ve seen so far. If both powers are all talk with nothing behind it and both powers know and understand this, then they’ll stick in the current high-tension equilibria or fight a war. The only stable states here are s. 0, 4, and 5. State 10, the “negotiated settlement” state is entirely absent. We’ll revisit this scenario with hypergame analysis later, to see what happens if the bluff is believed.

Symmetric Metarationality

Click for copyable version

Here we see more equilibria than we’ve seen in any of the other examples. States 2 (unilateral US withdrawal) and 8 (North Korea unilaterally abandoning its nuclear weapons program) make their debut and s. 0, 4, 5, 10, and 12 appear again.

Remember, Symmetric Metarationality is very risk averse; it considers not just opponents’ unilateral improvements, but all of their unilateral moves as fair game. The fact that s. 0 has unilateral moves for either side that are aggressive leaves the actors too scared to move to it, even from states that they disprefer. This explains the presence of s. 2 and s. 8 in the equilibrium for the first time; they’re here because in this model both sides are so scared of war that if they blink first, they’ll be more relieved at the end of tension than they will be annoyed at moving away from their preferences.

I think in general this is a poor assumption, which is why I tend to find Sequential Stability a more useful concept than Symmetric Metarationality. That said, I don’t think this is impossible as a state of affairs, so I’m glad that I observed it. In general, this is actually one of my favourite things about the Graph Model of Conflict Resolution: using it you can very quickly answer “what ifs”, often in ways that are easily bent to understandable narratives.

Why Sensitivity Analysis?

The cool thing about sensitivity analysis is that it shows you the equilibria a conflict can fall into and how sensitivity those equilibria are to your judgement calls. There are 12 possible states in this conflict, but only 7 of them showed up in any stability analysis at all. Within those seven, only 5 showed up more than once.

Here’s a full accounting of the states that showed up (counting our first model, there were nine possible simulations for each equilibrium to show up in):

State Explanation #
0 Status quo 7
2 Unilateral US troop withdrawal 1
4 North Korean invasion with only conventional US responses 4
5 North Korean invasion with US nuclear strike 9
8 Unilateral North Korean abandonment of nuclear weapons 1
10 Coordinated US withdrawal and NK abandonment of nuclear weapons 6
12 NK invasion after abandoning nuclear weapons; conventional US response 6

Of the five that showed up more than once, four showed up more than half the time. These then are the most robust equilibria; equilibria that half of the reasonable changes we attempted couldn’t dislodge.

Note “most robust” is not necessarily equivalent to “most likely”. To get actual probabilities on outcomes, we’d have to put probabilities on the initial conditions. Even then, the Graph Model of Conflict Resolution as we’ve currently talked about it does little to explain how decision makers move between equilibria; because this scenario starts in equilibrium, it’s hard to see how it makes it to any of the other equilibria.

Hopefully I’ll be able to explain one way we can model changes in states in my next post, which will cover Hypergame Analysis – the tool we use when actors lack a perfect understanding of one another’s preferences.

Model, Politics

The Graph Model of Conflict Resolution – Introduction

[10 minute read]

Why do things happen the way they do?

Every day, there are conflicts between decision makers. These occur on the international scale (think the Cuban Missile Crisis), the provincial level (Ontario’s sex-ed curriculum anyone?) and the local level (Toronto’s bike lane kerfuffle). Conflict is inevitable. Understanding it, regrettably, is not.

The final results of many conflicts can look baffling from the outside. Why did the Soviet Union retreat in the Cuban missile crisis? Why do some laws pass and others die on the table?

The most powerful tool I have for understanding the ebb and flow of conflict is the Graph Model of Conflict Resolution (GMCR). I had the immense pleasure of learning about it under the tutelage of Professor Keith Hipel, one of its creators. Over the next few weeks, I’d like to share it with you.

GMCR is done in two stages, modelling and analysis.

Modelling

To model a problem, there are four steps:

  • Select a point in time for the model
  • Make a list of the players and their options
  • Remove outcomes that don’t make sense
  • Create preference vectors for all players

The easiest way to understand this is to see it done.

Let’s look at the current nuclear stand-off on the Korean peninsula. I wrote this on Sunday, October 29th, 2017, so that’s the point in time we’ll use. To keep things from getting truly out of hand in our first example, let’s just focus on the US and North Korea (I’ll add in South Korea and China in a later post). What options does each side have?

US:

  • Nuclear strike on North Korea
  • Withdraw troops and normalize relations
  • Status quo

North Korea:

  • Invasion of South Korea
  • Abandon nuclear program and submit to inspections
  • Status quo

I went through a few iterations here. I originally wrote the US option “Nuclear strike” as “Pre-emptive strike”. I changed it to be more general. A nuclear strike could be pre-emptive, but it also could be in response to North Korea invading South Korea.

It’s pretty easy to make a chart of all these states:

Click for a copyable version

If you treat each action that the belligerents can make as a binary variable (yes=1 or no=0), the states will have a natural ordering based off of the binary sum of the actions taken and not taken. This specific ordering isn’t mandatory – you can use any ordering scheme you want – but I find it useful.

You may also notice that “Status quo” appears nowhere on this chart. That’s an interesting consequence of how actions are represented in the GMCR. Status quo is simply neither striking nor withdrawing for the US, or neither invading nor abandoning their nuclear program for North Korea. Adding an extra row for it would just result in us having to do more work in the next step, where we remove states that can’t exist.

I’ve colour coded some of the cells to help with this step. Removing nonsensical outcomes always requires a bit of judgement. Here we aren’t removing any outcomes that are highly dispreferred. We are supposed to restrict ourselves solely to removing outcomes that seem like they could never ever happen.

To that end, I’ve highlighted all cases where America withdraws troops and strikes North Korea. I’m interpreting “withdraw” here to mean more than just withdrawing troops – I think it would mean that the US would be withdrawing all forms of protection to South Korea. Given that, it wouldn’t make sense for the US to get involved in a nuclear war with North Korea while all the while loudly proclaiming that they don’t care what happens on the Korean peninsula. Not even Nixon’s “madman” diplomacy could encompass that.

On the other hand, I don’t think it’s necessarily impossible for North Korea to give up its nuclear weapons program and invade South Korea. There are a number of gambits where this might make sense – for example, it might believe that if they attacked South Korea after renouncing nuclear weapons, China might back them or the US would be unable to respond with nuclear missiles. Ultimately, I think these should be left in.

Here’s the revised state-space, with the twelve remaining states:

Click for a copyable version

The next step is to figure out how each decision maker prioritizes the states. I’ve found it’s helpful at this point to tag each state with a short plain language explanation.

State Explanation
0 Status quo
1 Nuclear strike by the US, NK keeps nuclear weapons
2 Unilateral US troop withdrawal
4 North Korean invasion with only conventional US responses
5 North Korean invasion with US nuclear strike
6 US withdrawal and North Korean Invasion
8 Unilateral North Korean abandonment of nuclear weapons
9 US strike and North Korean abandonment of nuclear weapons
10 Coordinated US withdrawal and NK abandonment of nuclear weapons
12 NK invasion after abandoning nuclear weapons; conventional US response
13 NK invasion after abandoning nuclear weapons; US nuclear strike
14 US withdrawal paired with NK nuclear weapons abandonment and invasion

While describing these, I’ve tried to avoid talking about causality. I didn’t describe s. 5 as “North Korean invasion in response to US nuclear strike” or “US nuclear strike in response to North Korean invasion”. Both of these are valid and would depend on which states preceded s. 5.

Looking at all of these states, here’s how I think both decision makers would order them (in order of most preferred to least preferred):

US: 8, 0, 9, 10, 12, 5, 4, 13, 14, 1, 2, 6
NK: 6, 14, 2, 10, 0, 4, 12, 5, 1, 13, 8, 9

The US prefers North Korea give up its nuclear program and wants to keep protecting South Korea. Its secondary objective is to seem like a reasonable actor on the world stage – which means that it has some preference against using pre-emptive strikes or nuclear weapons on non-nuclear states.

North Korea wants to unify the Korean peninsula under its banner, protect itself against regime change, and end the sanctions its nuclear program has brought. Based on the Agreed Framework, I do think Korea would be willing to give up nuclear weapons in exchange for a normalization of relations with the US and sanctions relief.

Once we have preference vectors, we’ve modelled the problem. Now it’s time for stability analysis.

Stability

A state is stable for a player if it isn’t advantageous for the player to shift states. A state is globally stable if it is not advantageous for any player to shift states. When a player can move to a state they prefer over the current state without any input from their opponent, this is a “unilateral improvement” (UI).

There are a variety of ways we can define “advantageous”, which lead to various definitions of stability:

Nash Stability (R): Stable if the actor has no unilateral improvements. States that are Nash stable tend to be pretty bad; these include both sides attacking in a nuclear war or both prisoners defecting in the prisoner’s dilemma. Nash stability ignores the concept of risk; it will never move to a less preferred state in the hopes of making it to a more preferred state.

General Metarationality (GMR): Stable if the actor has no unilateral improvements that aren’t sanctioned by unilateral moves by others. This tends to lead to less confusing results than Nash stability; Cooperation in the prisoner’s dilemma is stable in General Metarationality. General Metarationality accepts the existence of risk, but refuses to take any.

Symmetric Metarationality (SMR): Stable if an actor has no unilateral improvements that aren’t sanctioned by opponents’ unilateral moves after it has a chance to respond to them. This is equivalent to GMR, but with a chance to respond. Here we start to see the capacity to take on some risk.

Sequential Stability (SEQ): Stable if the actor has no unilateral improvements that aren’t sanctioned by opponents’ unilateral improvements. This basically assumes fairly reasonable opponents, the type who won’t cut off their nose to spite their face. Your mileage may vary as to how appropriate this assumption is. Like SMR, this system takes on some risk.

Limited Move Stability (LS): A state is stable if after N moves and countermoves (with both sides acting optimally), there exists no improvement. This is obviously fairly risky as any assumptions you make about your opponents’ optimal actions may turn out to be wrong (or wishful thinking).

Non-myopic Stability (NM): Equivalent to Ls with N set equal to infinity. This predicts stable states where there’s no improvements after any amount of posturing and state changes, as long as both players act entirely optimally.

The two stability metrics most important to the GMCR (at least as I was taught it) are Nash Stability (denoted with r) and Sequential Stability (denoted with s). These have the advantage of being simple enough to calculate by hand while still explaining most real-world equilibria quite well.

To do stability analysis, you write out the preference vectors of both sides, along with any unilateral improvements that they can make. You then use this to decide the stability of each state for each player. If both players are stable at a state by any of the chosen stability metrics, the state overall is stable. A state can also be stable if both players have unilateral improvements from it that result in both ending up in a dispreferred state if taken simultaneously. This is called simultaneous sanctioning and is denoted with u.

The choice of stability metrics will determine which states are stable. If you only use Nash stability, you’ll get a different result than if you combine Sequential Stability and Nash Stability.

Here’s the stability analysis for this conflict (using Nash Stability and Sequential Stability):

Click for copyable version

Before talking about the outcome, I want to mention a few things.

Look at s. 9 for the US. They prefer s. 8 to s. 9 and the two differ only on a US move. Despite this, s. 8 isn’t a unilateral improvement over s. 9 for the US. This system is called the Graph Model of Conflict Resolution for a reason. States can be viewed as nodes on a directed graph, which implies that some nodes may not have a connection. Or, to put it in simpler terms, some actions can’t be taken back. Once the US has launched a nuclear strike, it cannot un-launch it.

This holds less true for abandoning a nuclear program or withdrawing troops; both of those are fairly easy to undo (as we found out after the collapse of the Agreed Framework). Invasions on the other hand are in a tricky category. They’re somewhat reversible (you can stop and pull out), but the consequences linger. Ultimately I’ll call them reversible, but note that this is debatable and the analysis could change if you change this assumption.

In a perfect world, I’d go through this exercise four or five different times, each time with different assumptions about preferences or the reversibility of certain states or with different stability metrics and see how each factor changes the results. My next blog post will go through this in detail.

The other thing to note here is the existence of simultaneous sanctioning. Both sides have a UI from s. 4; NK to s. 0 and the US to s. 5. Unfortunately, if you take these together, you get s. 1, which both sides disprefer to s. 4. This means that once a war starts the US will be hesitant to launch a nuclear strike and North Korea would be hesitant to withdraw – in case they withdrew just as a strike happened. In reality, we get around double binds like this with negotiated truces – or unilateral ultimatums (e.g. “withdraw by 08:00 tomorrow or we will use nuclear weapons”).

There are four stable equilibria in this conflict:

  • The status quo
  • A coordinated US withdrawal of troops (but not a complete withdrawal of US interest) and North Korean renouncement of nuclear weapons
  • All out conventional war on the Korean Peninsula
  • All out nuclear war on the Korean Peninsula

I don’t think these equilibria are particularly controversial. The status quo has held for a long time, which would be impossible if it wasn’t a stable equilibrium. Meanwhile, s. 10 looks kind of similar to the Iran deal, with the US removing sanctions and doing some amount of normalization in exchange for the end of Iran’s nuclear program. State 5 is the worst-case scenario that we all know is possible.

Because we’re currently in a stable state, it seems unlikely that we’ll shift to one of the other states that could exist. In actuality, there are a few ways this could happen. A third party could intervene with its own preference vectors and shake up the equilibrium. For example, China could use the threat of economic sanctions (or the threat of ending economic sanctions) to try and get North Korea and the US to come to a détente. There also could be an error in judgement on the part of one of the parties. A false alarm could quickly turn into a very real conflict. It’s also possible that one party could mistake the others preferences, leading to them taking a course of action that they incorrectly believe isn’t sanctioned.

In future posts, I plan to show how these can all be taken into account, using the GMCR framework for Third Party Intervention and Coalitional Analysis, Strength of Preferences, and Hypergame Analysis.

Even without those additions, the GMCR is a powerful tool. I encourage you to try it out for other conflicts and see what the results are. I certainly found that the best way to really understand it was to run it a few times.

Note: I know it’s hard to play around with the charts when they’re embedded as images. You can see copyable versions of them here.